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Hindawi Publishing CorporationFixed Point Theory and Applications Volume 2011, Article ID 840978, 12 pages doi:10.1155/2011/840978 Research Article Algorithm for General Nonlinear Mixed

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Hindawi Publishing Corporation

Fixed Point Theory and Applications

Volume 2011, Article ID 840978, 12 pages

doi:10.1155/2011/840978

Research Article

Algorithm for General Nonlinear Mixed Set-Valued Inclusion Framework

Xian Bing Pan,1 Hong Gang Li,2 and An Jian Xu3

1 Yitong College, Chongqing University of Posts and Telecommunications, Chongqing 400065, China

2 Institute of Applied Mathematics Research, Chongqing University of Posts and Telecommunications, Chongqing 400065, China

3 College of Mathematics and Statistics, Chongqing University of Technology, Chongqing 400054, China

Correspondence should be addressed to Xian Bing Pan,panxianb@163.com

Received 16 November 2010; Accepted 10 January 2011

Academic Editor: T Benavides

Copyrightq 2011 Xian Bing Pan et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

The purpose of this paper is1 a general nonlinear mixed set-valued inclusion framework for

the over-relaxed A-proximal point algorithm based on the A, η-accretive mapping is introduced,

and2 it is applied to the approximation solvability of a general class of inclusions problems using the generalized resolvent operator technique due to Lan-Cho-Verma, and the convergence of

iterative sequences generated by the algorithm is discussed in q-uniformly smooth Banach spaces.

The results presented in the paper improve and extend some known results in the literature

1 Introduction

In recent years, various set-valued variational inclusion frameworks, which have wide applications to many fields including, for example, mechanics, physics, optimization and control, nonlinear programming, economics, and engineering sciences have been intensively studied by Ding and Luo1, Verma 2, Huang 3, Fang and Huang 4, Fang et al 5, Lan et al 6, Zhang et al 7, respectively Recently, Verma 8 has intended to develop

a general inclusion framework for the over-relaxed A-proximal point algorithm 9 based

on the A-maximal monotonicity In 2007-2008, Li 10,11 has studied the algorithm for a new class of generalized nonlinear fuzzy set-valued variational inclusions involvingH,

η-monotone mappings and an existence theorem of solutions for the variational inclusions, and a new iterative algorithm 12 for a new class of general nonlinear fuzzy mulitvalued quasivariational inclusions involving G, η-monotone mappings in Hilbert spaces, and

discussed a new perturbed Ishikawa iterative algorithm for nonlinear mixed set-valued

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quasivariational inclusions involvingA, η-accretive mappings, the stability 13 and the

convergence of the iterative sequences in q-uniformly smooth Banach spaces by using the

resolvent operator technique due to Lan et al.6

Inspired and motivated by recent research work in this field, in this paper, a general

nonlinear mixed set-valued inclusion framework for the over-relaxed A-proximal point

algorithm based on the A, η-accretive mapping is introduced, which is applied to the

approximation solvability of a general class of inclusions problems by the generalized resolvent operator technique, and the convergence of iterative sequences generated by

the algorithm is discussed in q-uniformly smooth Banach spaces For more literature, we

recommend to the reader1 17

2 Preliminaries

Let X be a real Banach space with dual space X∗, and let ·, · be the dual pair between X and X∗, let 2X denote the family of all the nonempty subsets of X, and let CBX denote the family of all nonempty closed bounded subsets of X The generalized duality mapping

J q : X → 2 X∗is single-valued if X∗is strictly convex14, or X is uniformly smooth space In what follows we always denote the single-valued generalized duality mapping by J qin real

uniformly smooth Banach space X unless otherwise stated We consider the following general nonlinear mixed set-valued inclusion problem with A, η-accretive mappings (GNMSVIP).

Finding x ∈ X such that

0∈ FAx Mx, 2.1

where A, F : X → X, η : X × X → X be single-valued mappings; M : X → 2 X be an

A, η-accretive set-valued mapping A special case of problem 2.1 is the following:

if X X∗ is a Hilbert space, F 0 is the zero operator in X, and ηx, y x − y,

then problem2.1 becomes the inclusion problem 0 ∈ Mx with a A-maximal monotone mapping M, which was studied by Verma 8

Definition 2.1 Let X be a real Banach space with dual space X∗, and let·, · be the dual pair between X and X∗ Let A : X → X and η : X × X → X be single-valued mappings A set-valued mapping M : X → 2 Xis said to be

i r−strongly η-accretive, if there exists a constant r > 0 such that

y1− y2, J q

ηx1, x2≥ rx1− x2q

, ∀y i ∈ Mx i , i 1, 2; 2.2

ii m-relaxed η-accretive, if there exists a constant m > 0 such that

y1− y2, J q

ηx1, x2≥ −mx1− x2q , ∀x1, x2∈ X, y i ∈ Mx i , i 1, 2; 2.3

iii c-cocoercive, if there exists a constant c such that

y1− y2, J q

ηx1, x2≥ cy1− y2q

, ∀x1, x2∈ X, y i ∈ Mx i , i 1, 2; 2.4

iv A, η-accretive, if M is m-relaxed η-accretive and RA ρMX X for every

ρ > 0.

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Fixed Point Theory and Applications 3

Based on the literature6, we can define the resolvent operator R A,η ρ,Mas follows

Definition 2.2 Let η : X × X → X be a single-valued mapping, A : X → X be a strictly η-accretive single-valued mapping and M : X → 2 X be an A, η-accretive set-valued mapping The resolvent operator R A,η ρ,M : X → X is defined by

R A,η ρ,M x A ρM−1

x ∀x ∈ X, 2.5

where ρ > 0 is a constant.

Remark 2.3 The A, η-accretive mappings are more general than H, η-monotone mappings and m-accretive mappings in Banach space or Hilbert space, and the resolvent operators

associated with A, η-accretive mappings include as special cases the corresponding

resolvent operators associated with H, η-monotone operators, m-accretive mappings, A-monotone operators, η-subdiﬀerential operators 1 7,11–13

Lemma 2.4 see 6 Let η : X × X → X be τ-Lipschtiz continuous mapping, A : X → X be an r-strongly η-accretive mapping, and M : X → 2 X be an A, η-accretive set-valued mapping Then the generalized resolvent operator R A,η ρ,M : X → X is τ q−1 /r − mρ-Lipschitz continuous, that is,

R A,η ρ,M x − R A,η

ρ,M

y ≤ τ q−1

r − mρx − y ∀x,y ∈ X, 2.6

where ρ ∈ 0, r/m.

In the study of characteristic inequalities in q-uniformly smooth Banach spaces, Xu

14 proved the following result

Lemma 2.5 Let X be a real uniformly smooth Banach space Then X is q-uniformly smooth if and

only if there exists a constant c q > 0 such that for all x, y ∈ X,

x yq

≤ x q qy, J q x c qyq

This section deals with an introduction of a generalized version of the over-relaxed proximal point algorithm and its applications to approximation solvability of the inclusion problem of the form2.1 based on the A, η-accretive set-valued mapping.

Let M : X → 2 X be a set-valued mapping, the set{x, y : y ∈ Mx} be the graph

of M, which is denoted by M for simplicity, This is equivalent to stating that a mapping is any subset M of X × X, and Mx {y : x, y ∈ M} If M is single-valued, we shall still use

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Mx to represent the unique y such that x, y ∈ M rather than the singleton set {y} This interpretation will depend greatly on the context The inverse M−1of M is {y, x : x, y ∈ M}.

Definition 3.1 Let M : X → 2 X be a set-valued mapping The map M−1, the inverse of

M : X → 2 X, is said to be generalu, t-Lipschitz continuous at 0 if, and only if there exist two constants u, t ≥ 0 for any w ∈ B t {w : w ≤ t, w ∈ X}, a solution x∗ of the inclusion

0∈ Mxx∗∈ M−10 exist and the x∗such that

x − x∗ ≤ uw ∀x ∈ M−1w, 3.1

holds

Lemma 3.2 Let X be a q-uniformly smooth Banach space, η : X × X → X be a τ-Lipschtiz

continuous mapping, A : X → X be an r-strongly η-accretive mapping, F : X → X be a ξ-Lipschtiz continuous mapping, and M : X → 2 X be an A, η-accretive set-valued mapping If

I k A − AR A,η

ρ,M A − ρFA, and for all x1, x2∈ X, ρ > 0 and qγ > 1

Ax1 − Ax2, J q

A

R A,η ρ,M

Ax1 − ρFAx1− AR A,η ρ,M

Ax2 − ρFAx2

≥ γA

R A,η ρ,M

Ax2 − ρFAx2q

,

3.2

then

qγ − 1AR A,η

ρ,M

Ax2 − ρFAx2q

 I k x1 − I k x2q ≤ c q Ax1 − Ax2q

.

3.3

Proof Let X be a q-uniformly smooth Banach space, η : X × X → X be a τ-Lipschtiz continuous mapping, A : X → X be an r-strongly η-accretive mapping, and M : X → 2 X

be an A, η-accretive set-valued mapping Let us set I k A − AR A,η

ρ,M A − ρFA and

s i Ax i − ρFAx i x i ∈ X, i 1, 2, then by usingDefinition 2.2, Lemmas2.4,2.5, and

3.2, we can have

I k x1 − I k x2q

Ax

1 − AR A,η ρ,M s1−Ax2 − AR A,η ρ,M s2q

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≤ c q Ax1 − Ax2q − q Ax1 − Ax2, J q

A

R A,η ρ,M s1− AR A,η ρ,M s2

A

R A,η ρ,M

Ax2 − ρFAx2q

≤ c q Ax1 − Ax2q

− qγA

R A,η ρ,M

Ax2 − ρFAx2q

A

R A,η ρ,M

Ax2 − ρFAx2q

≤ c q Ax1 − Ax2q

−qγ − 1AR A,η

ρ,M

Ax2 − ρFAx2q

.

3.4 Therefore,3.3 holds

continuous mapping, A : X → X be an r-strongly η-accretive and nonexpansive mapping,

F : X → X be an ξ-Lipschtiz continuous mapping, and I k A − AR A,η ρ,M A − ρFA, and

M : X → 2 X be an A, η-accretive set-valued mapping Then the following statements are mutually equivalent.

i An element x∗∈ X is a solution of problem 2.1.

ii For a x∗∈ X, such that

x∗ R A,η ρ,M

Ax∗ − ρFAx∗. 3.5

iii For a x∗∈ X, holds

I k x∗  Ax∗ − AR A,η ρ,M

Ax∗ − ρFAx∗ 0, 3.6

where ρ > 0 is a constant.

Proof This directly follows from definitions of R A,η ρ,Mx and I k

continuous mapping, A : X → X be an r-strongly η-accretive and nonexpansive mapping, F : X →

X be an ξ-Lipschtiz continuous and β-strongly η-accretive mapping, and I k A − AR A,η

ρ,M A − ρFA, and M : X → 2 X be an A, η-accretive set-valued mapping If the following conditions holds

τ q q 1 c q ρ q ξ q − qρβ < τr − mρ

1 c q ρ q ξ q > qρβ

, 3.7

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where c q > 0 is the same as in Lemma 2.5 , and ρ ∈ 0, r/m Then the problem 2.1 has a solution

x∗∈ X.

Proof Define N : X → X as follows:

Nx R A,η

ρ,M

Ax − ρFAx, ∀x ∈ X. 3.8

For elements x1, x2 ∈ X, if letting

s i Ax i − ρFAx i i 1, 2, 3.9 then by3.1 and 3.3, we have

Nx1 − Nx2 R A,η

ρ,M s1 − R A,η

ρ,M s2

≤ τ q−1

r − mρAx1 − Ax2 − ρFAx1 − FAx2. 3.10

By using r-strongly η-accretive of A, β-strongly η-accretive of F, andLemma 2.5, we obtain

Ax1 − Ax2 − ρFAx1 − FAx2q

≤ Ax1 − Ax2q c q ρ q FAx1 − FAx2q

− qρFAx1 − FAx2, J q Ax1 − Ax2

≤1 c q ρ q ξ q − qρβAx1 − Ax2q

3.11

Combining3.10-3.11, by using nonexpansivity of A, we have

Nx1 − Nx2 ≤ θ∗x1− x2, 3.12 where

θ∗ τ q−1

r − mρ

q1 c q ρ q ξ q − qρβ 1 c q ρ q ξ q > qρβ

.

3.13

It follows from3.7–3.12 that N has a fixed point in X, that is, there exist a point x∗ ∈ X such that x∗ Nx∗, and

x∗ Nx∗  R A,η

ρ,M

Ax∗ − ρFAx∗. 3.14 This completes the proof

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Based on Lemma 3.3, we can develop a general over-relaxed A, η-proximal point

algorithm to approximating solution of problem2.1 as follows

Algorithm 3.5 Let X be a q-uniformly smooth Banach space, η : X × X → X be a τ-Lipschtiz continuous mapping, A : X → X be an r-strongly η-accretive and nonexpansive mapping,

F : X → X be an β-strongly η-accretive mapping and ξ-Lipschitz continuous, and I k

A − AR A,η ρ,M A − ρFA, and M : X → 2 X be an A, η-accretive set-valued mapping Let {a n}∞n0 a n ≥ 1, {b n}∞n0and{ρ n}∞n0be three nonnegative sequences such that

∞

n1

b n < ∞, a lim sup

n → ∞ a n ≥ 1, ρ n ↑ ρ ≤ ∞, 3.15

where ρ n , ρ ∈ 0, r/mn 0, 1, 2, ·, ·, · and each satisfies condition 3.7

Step 1 For an arbitrarily chosen initial point x0∈ X, set

Ax1 1 − a0Ax0 a0y0, 3.16

where the y0satisfies

y0− AR A,η ρ0,M

Ax0 − ρ0FAx0 ≤ b

0y0− Ax0. 3.17

Step 2 The sequence {x n} is generated by an iterative procedure

Ax n1 1 − a n Ax n a n y n , 3.18

and y nsatisfies

y n − AR A,η ρ n ,M

Ax n − ρ n FAx n ≤ b

ny n − Ax n, 3.19

where n 1, 2, ·, ·, ·.

Remark 3.6 For a suitable choice of the mappings A, η, F, M, I k , and space X, then the

over-relaxed A-proximal point algorithm 8

Theorem 3.7 Let X be a q-uniformly smooth Banach space Let A, F : X → X and η : X × X → X

be single-valued mappings, and let M : X × X → 2 X be a set-valued mapping and FA M−1be the inverse mapping of the mapping FA M : X → 2 X satisfying the following conditions:

i η : X × X → X is τ-Lipschtiz continuous;

ii A : X → X be an r-strongly η-accretive mapping and nonexpansive;

iii F : X → X be an ξ-Lipschtiz continuous and β-strongly η-accretive mapping;

iv M : X → 2 X be an A, η-accretive set-valued mapping;

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v the FA M−1be u, t-Lipschitz continuous at 0u ≥ 0;

vi {a n}∞

n0 a n ≥ 1, {b n}∞

n0 and {ρ n}∞

n0 be three nonnegative sequences such that

∞

n1

b n < ∞, a lim sup

n → ∞ a n ≥ 1, ρ n ↑ ρ ≤ ∞, 3.20

where ρ n , ρ ∈ 0, r/mn 0, 1, 2, ·, ·, · and each satisfies condition 3.7,

vii let the sequence {x n } generated by the general over-relaxed A-proximal point algorithm

3.6 be bounded and x∗be a solution of problem2.1, and the condition

Ax n − Ax∗, J q

A

R A,η ρ,M

Ax n − ρFAx n− AR A,η ρ,M

Ax∗ − ρFAx∗

≥ γA

R A,η ρ,M

Ax n − ρFAx n− AR A,η ρ,M

Ax∗ − ρFAx∗q

,

3.21

0 < c q a − 1 qa q − qa − 1aγd q < 1, 3.22

hold Then the sequence {x n } converges linearly to a solution x∗ of problem2.1 with convergence rate ϑ, where

ϑ q

c q a − 1 qa q q1 − aaγd q ,

a lim sup

n → ∞

a n , d lim sup

n → ∞

d n lim sup

n → ∞

q

c q u q

qγ − 1

r q u q ρ q

n

,

∞

n1

b n < ∞.

3.23

Proof Let the x∗ be a solution of the Framework 2.1 for the conditions i–iv and

Ax∗  1 − a n Ax∗  a n A

R A,η ρ n ,M

Ax∗ − ρ n FAx∗. 3.24

We infer fromLemma 3.3that any solution to2.1 is a fixed point of R A,η

ρ n ,M A − ρ n FA First,

in the light ofLemma 3.2, we show

R A,η ρ n ,MAx n − ρ n FAx n− x∗ ≤ d

n Ax n − Ax∗, 3.25

where d n q c q u q /2γ − 1r q u q ρ q

n < 1 and R A,η

ρ ,M Ax∗ − ρ n FAx∗  x∗

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For I k A − AR A,η ρ,M A − ρ n FA, and under the assumptions including the condition

vii 3.21, then I k x n → 0n → ∞ since the FAM−1isu, t-Lipschitz continuous at 0 Indeed, it follows that R A,η ρ n ,M Ax n − ρ n FAx n ∈ FA M−1ρ−1

n I k x n from ρ−1

n I k x n ∈

FAMR A,η

ρ n ,M Ax n −ρ n FAx n Next, by using the condition iv and 3.1, and setting

w ρ−1n I k x n and z R A,η

ρ n ,M Ax n − ρ n FAx n, we have

R A,η

ρ n ,M

Ax n − ρ n FAx n− x∗ ≤ uρ−1

n I k x n, ∀n > n . 3.26

Now applyingLemma 3.3, we get

R A,η ρ n ,MAx n − ρ n FAx n− x∗q

≤R A,η

ρ n ,M

Ax n − ρ n FAx n− R A,η

ρ n ,M

Ax∗ − ρ n FAx∗q

≤ u qρ−1

n I k x n − ρ−1

n I k x∗q

≤

u

ρ n

q

I k x n − I k x∗q

≤

u

ρ n

q

−qγ − 1

r qR A,η

ρ n ,M

Ax n − ρ n FAx n− R A,η

ρ n ,M

Ax∗ − ρ n FAx∗q

c q Ax n − Ax∗q

.

3.27

Therefore,

R A,η

ρ n ,M

Ax n − ρ n FAx n− x∗ ≤ d

n Ax n − Ax∗, 3.28

where d n q

c q u q /2γ − 1r q u q ρ q

n < 1 and R A,η

ρ n ,M Ax∗ − ρ n FAx∗  x∗ Next we start the main part of the proof by using the expression

Az n1 1 − a n Ax n a n A

R A,η ρ ,M

Ax n − ρ n FAx n, ∀n ≥ 0. 3.29

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Let us set s n Ax n − ρ n FAx n and s∗  Ax∗ − ρ n FAx∗ for simple We begin with estimatingfor a n ≥ 1 and later using 3.2, the nonexpansivity of A, 3.21 and 3.28 as follows:

Az n1 − Ax∗q ≤1 − a

n Ax n a n A

R A,η ρ n ,M s n

−1 − a n Ax∗  a n A

R A,η ρ n ,M s∗q

≤1 − a

n Ax n − Ax∗  a n

A

R A,η ρ n ,M s n− AR A,η ρ n ,M s∗q

≤ c q 1 − a n Ax n − Ax∗qa

n

A

R A,η ρ n ,M s n− AR A,η ρ n ,M s∗q

 q1 − a n a n

Ax n − Ax∗, J q

A

R A,η ρ n ,M s n− AR A,η ρ n ,M s∗

≤ c q a n− 1q Ax n − Ax∗q a q

nR A,η

ρ n ,M s n − R A,η

ρ n ,M s∗q

 q1 − a n a n γR A,η

ρ n ,M s n − R A,η

ρ n ,M s∗q

≤ c q a n− 1q Ax n − Ax∗q

a q n − q1 − a n a n γR A,η

ρ n ,M

Ax n − ρ n FAx n− x∗q

≤c q a n− 1qa q n q1 − a n a n γ

d q n

Ax n − Ax∗q

.

3.30 Thus, we have

Az n1 − Ax∗q ≤ θ n Ax n − Ax∗q , 3.31 where

θ n q

c q a n− 1qa q n q1 − a n a n γ

d q n < 1, 3.32

and a q n q1 − a n a n γ > 0, a n≥ 1,∞n1 b n < ∞, and d n q

c q u q /2γ − 1r q u q ρ q n < 1 Since Ax n1 1 − a n Ax n a n y n , we have Ax n1 − Ax n a n y n − Ax n It follows that

Ax n1 − Az n1

≤1 − a

n Ax n a n y n−1 − a n Ax n a n R A,η ρ n ,M

Ax n − ρ n FAx n

≤ a ny

n − R A,η

ρ n ,M

Ax n − ρ n FAx n

≤ a n b ny n − Ax n.

3.33

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where c q > is the same as in Lemma 2.5 , and ρ ∈ 0, r/m Then the problem 2.1... ρFAx∗. 3.14 This completes the proof

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Based...

Remark 3.6 For a suitable choice of the mappings A, η, F, M, I k , and space X, then the< /i>

over-relaxed A-proximal point algorithm 8

Theorem 3.7

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